Is $\cos(x^2)$ the same as $\cos^2(x)$?

Question

I want to know something about trigonometrical functions, is $\cos(x^2)$ the same as $\cos^2(x)$ ?

$\cos^2(x) = \cos(x)\times \cos(x)$ and $\cos(x^2) = \cos(x \times x)$ So no. But beware, the notation $\cos^{-1}(x)$ is ambiguous. It can denote the inverse cosine function or the reciprocal of the cosine function. — Nigel Overmars, Jan 27 '14 at 11:44
When you have a doubt like this and you are lazy to find a formal proof that concludes that your hyphotesis is true, you can probe a few random values until you find a counterexample that invalidate the hyphotesis of your question. For example, is $cos (33^2) = cos^2(33)$? No? Then you can properly conclude that the answer of your question is false :) — Daniel Muñoz Parsapoormoghadam, Jan 27 '14 at 12:19

score 4 · Answer 1 · 2014-01-27T12:06:37.487

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One of the first things you may observe is that $(\cos {\small{(}}x{\small{)}})^2\geqslant0$ whereas $\cos(x^2)$ may be equal to a negative number. (Why?)

In blue, a graph of the function $\color{blue}{\cos^2(x)}$ and in red a graph of the function $\color{red}{\cos(x^2)}$.

made using GeoGebra

Some say, a good plot is worth a million words! :-)

edited Jan 27 '14 at 12:06

answered Jan 27 '14 at 11:50

AlexR · Accepted Answer · 2014-01-27T11:52:40.970

For a general function $f$, which can be about anything and is $\cos$ in your case and with $g(x)=x^n$, $$f^n(x) := (f(x))^n = g\circ f(x) = g(f(x))$$ and $$f(x^n) := f(\underbrace{x\cdot x \cdot \ldots}_{n \text{ times}}) = f \circ g(x) = f(g(x))$$ are two different functions.
Note for trigonometric functions, $\cos^{-1}$ sometimes refers to $\arccos$, and sometimes to $\sec = \frac1{\cos}$, so you should be careful about exponentiating functions.

score 2 · Answer 3 · answered Jan 28 '15 at 15:38

There are few functions such that $f^2(x)=f(x^2)$: essentially the powers of $x$, $f(x)=x^n$: $f^2(x)=(x^n)^2=(x^2)^n=f(x^2)$.

The rule is more often $f^2(x)\ne f(x^2)$. Just an example with$f(x)=x+1$: $(x+1)^2\ne x^2+1$.

score 0 · Answer 4 · edited Dec 02 '19 at 19:05

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No. $\cos x^2=\cos(x^2)=\cos(x\times x)$ while $\cos^2(x)=\cos(x)\times \cos(x)=(\cos(x))^2$.

edited Dec 02 '19 at 19:05

Kenta S

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answered Jan 28 '15 at 15:31

SQL Injection

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why is my answer downvoted but yvesdaoust answer isn't? – SQL Injection Jan 28 '15 at 15:55
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Yves's answer isn't repeating what was said a year ago, but adding additional useful information. – Jonas Meyer Jan 28 '15 at 17:27

score 0 · Answer 5 · edited Jan 27 '14 at 11:49

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No $\cos^2(x)$ means $(\cos x)^2$. This is not the same as $\cos(x^2)$.

edited Jan 27 '14 at 11:49

jimjim

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answered Jan 27 '14 at 11:45

J.R.

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Why $(cos x)^{2}$ is not the same as $cos(x^2)$? – cosineofx Jan 27 '14 at 11:54
Just plug in a value, e.g. $x=\pi/2$ (that is 90 degrees), then $(\cos(x))^2=0^2=0$, but $\cos(\pi^2/4)<0$ because $\pi/2<\pi^2/4<\pi$. – J.R. Jan 27 '14 at 11:58

Is $\cos(x^2)$ the same as $\cos^2(x)$?

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